Question

rolling a car down a ramp
height (centimeters) | distance rolled (centimeters)
test 1 | 5 cm | 16 cm
test 2 | ? | 22 cm
test 3 | 15 cm | 27 cm
25 cm
2 cm
10 cm
30 cm

Explanation:

Step1: Analyze the relationship

We assume a linear relationship between height (h) and distance rolled (d), so we can use the formula for a line \( d = mh + b \). Using Test 1 (\( h_1 = 5, d_1 = 16 \)) and Test 3 (\( h_3 = 15, d_3 = 27 \)) to find the slope \( m \).
\( m=\frac{d_3 - d_1}{h_3 - h_1}=\frac{27 - 16}{15 - 5}=\frac{11}{10} = 1.1 \)

Step2: Find the y-intercept \( b \)

Using Test 1: \( 16 = 1.1\times5 + b \)
\( 16 = 5.5 + b \)
\( b = 16 - 5.5 = 10.5 \)
So the equation is \( d = 1.1h + 10.5 \)

Step3: Solve for \( h \) when \( d = 22 \)

\( 22 = 1.1h + 10.5 \)
\( 1.1h = 22 - 10.5 \)
\( 1.1h = 11.5 \) Wait, no, wait, maybe a better way: check the difference. From Test1 (5,16) to Test3 (15,27): height increases by 10, distance by 11. So per 10 height, distance +11. So from Test1 to Test2: distance +6 (22-16=6). So height increase: (6/11)*10? No, maybe my first approach was wrong. Wait, maybe it's a linear relationship, so let's use two points (5,16) and (15,27). The slope is (27-16)/(15-5)=11/10=1.1. Then equation is d=1.1h + b. Plug in (5,16): 16=5.5 + b => b=10.5. Then for d=22: 22=1.1h +10.5 => 1.1h=11.5 => h=11.5/1.1≈10.45, which is approximately 10. So the answer should be 10 cm.
Wait, maybe the intended approach is to see the pattern. Test1: 5,16; Test3:15,27. The height increases by 10 (15-5), distance by 11 (27-16). So the rate is 11 distance per 10 height. Now Test2 distance is 22, which is 16 +6. So 6 is 6/11 of 11, so height increase is 6/11 *10? No, maybe the problem is designed as a linear relationship with integer values. Let's check the options: 25,2,10,30. 10 is an option. Let's check: if h=10, then d=? Let's see the difference between Test1 (5,16) and Test2 (10,22): height +5, distance +6. Then Test2 (10,22) to Test3 (15,27): height +5, distance +5. Wait, no, that's not linear. Wait, maybe the problem has a typo, but the options include 10. So the answer is 10 cm.

Answer:

10 cm